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As you’re probably aware, I’m a big believer in using stories to bring math to life. Especially when you’re teaching tricky concepts, using a story can be the “magic switch” that flicks on the light of understanding. Armed with story-based understanding, students can recall how to perform difficult math processes. And since people naturally like stories and tend to recall them, skills based on story-based understanding really stick in the mind. I’ve seen this over and over in my tutoring.

The kind of story I’m talking about uses an extended-metaphor, and this way of teaching is particularly helpful when you’re teaching algebra. Ask yourself: what would you rather have? Students scratching their heads (or tearing out their hair) to grasp a process taught as a collection of abstract steps? Or students grasping a story and quickly seeing how it guides them in doing the math? I think the answer is probably pretty clear. So with this benefit in mind, let’s explore another story that teaches a critical algebraic skill: the skill of “unpacking” terms locked inside parentheses.

To get the picture, first imagine that each set of parentheses, weirdly or not, represents a corrugated cardboard box, the kind that moving companies use to pack up your possessions. Extending this concept, the terms inside parentheses represent the items you pack when you move your goodies from one house to another. Finally, for every set of parentheses (the box), imagine that you’ve hired either a good moving company or a bad moving company. (You can use a good company for one box and a bad company for a different “box” — it changes.) How can you tell whether the moving company is good or bad? Just look at the sign to the left of the parentheses. If the moving company is GOOD, you’ll see a positive sign to the left of the parentheses. If the moving company is BAD, you’ll spot a negative sign there.

Here’s how this idea looks:

+ ( ) The + sign here means you’ve hired a GOOD moving company for this box of stuff.

– ( ) This – sign means that you’ve hired a BAD moving company to pack up this box of things.

Now let’s put a few “possessions” inside the boxes.

+ (2x – 4) This means a GOOD moving company has packed up your treasured items: the 2x and the – 4.

– (2x – 4) Au contraire! This means that a BAD moving company has packed up the 2x and the – 4.

[Remember, of course, that the term 2x is actually a + 2x. No sign visible means there’s an invisible + sign before the term.]

What difference does it make if the moving company is GOOD or BAD? A big difference! If it’s a GOOD company, it packs your things up WELL. Result: when you unpack your items, they come out exactly the same way in which they went into the box. So since a good moving company packed up your things in the expression: + (2x – 4), when you go to unpack your things, everything will come out exactly as it went in. Here’s a representation of this unpacking process:

+ (2x – 4)

= + 2x – 4

Note that when we take terms out of parentheses, we call this “unpacking” the terms. This works because algebra teachers fairly often describe the process of taking terms out of ( ) as “unpacking” the terms. So here’s a story whose rhetoric matches the rhetoric of the algebraic process. Convenient, is it not?

Now let’s take a look at the opposite situation — what happens when you work with a BAD (boo, hiss!) moving company. In this case, the company does such a bad job that when you unpack your items, each and every item comes out “broken.” In math, we indicate that terms are “broken” by showing that when they come out of the ( ), their signs, + or – signs, are the EXACT OPPOSITE of what they should be. So if a term was packed up as a + term, it would come out as a – term. Vice-versa, if it was packed up as a – term, it would come out as a + term. We show the process of unpacking terms packed by a BAD moving company, as follows:

– (2x – 4)

= – 2x + 4

And that pretty much sums up the entire process. Understanding this story, students will be able to “unpack” terms from parentheses, over and over, with accuracy and understanding.

But since Practice Makes Perfect, here are a few problems to help your kiddos perfect this skill.

PROBLEMS:

“Unpack” these terms by removing the parentheses and writing the terms’ signs correctly:

a) – (5a + 3)

b) + (5a – 3)

c) – (– 3a + 2b – 7)

d) + (– 3a + 2b – 7)

e) 6 + (3a – 2)

f) 6 – (3a – 2)

g) 4a + 6 + (– 9a – 5)

h) 4a + 6 – (– 9a – 5)

ANSWERS:

a) – (5a + 3) = – 5a – 3

b) + (5a – 3) = + 5a – 3

c) – (– 3a + 2b – 7) = + 3a – 2b + 7

d) + (– 3a + 2b – 7) = – 3a + 2b – 7

e) 6 + (3a – 2) = + 3a + 4

f) 6 – (3a – 2) = – 3a + 8

g) 4a + 6 + (– 9a – 5) = – 5a + 1

h) 4a + 6 – (– 9a – 5) = + 13a + 11

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!

But when we teach math, we often expect something similar from students. We expect them to learn a complex, multi-step process in one lesson, in one hour. We expect them to go from no awareness of the process, to awareness to competence to mastery. And we don’t take account of the fact that many math process requires a long ladder of thought steps. In edu-jargon, this process of taking all of the little steps into account — and teaching each step individually — is called “scaffolding.”

Like climbing Everest, doing Math requires many STEPS

I have long found “scaffolding” important in working with students who struggle with math in general and algebra in particular. (more…)

This is a simple trick that anyone can easily learn. It is just a trick for
multiplying a number by 25.

If someone asked you what 25 times 36 equals, you’d probably be tempted
to reach for a calculator and start punching buttons. But remarkably, you’d
probably be able to work it out even faster in your head.

Since 25 is one-fourth of 100, multiplying by 25 is the same thing as
multiplying by 100 and dividing by 4. Or, even more simply:
first divide by 4,
then add two zeros.

Here’s the example:

Problem: 36 x 25
First divide 36 by 4 to get 9.
Then add two zeros to get: 900.
That, amazingly enough, is the answer.

Another example: 88 x 25
First divide 88 by 4 to get 22.
Then add two zeros to get: 2,200.

But, you say, what if the number you start with is not divisible by 4.
No problem. Just use this fact:
if the remainder is 1, that is the same as 1/4 or .25
if the remainder is 2, that is the same as 2/4 or .50
if the remainder is 3, that is the same as 3/4 or .75

So take a problem like this: 25 x 17
dividing 17 by 4, you get 4 remainder 1.
But that is the same as 4.25
Now just move the decimal right two places (same as multiplying by 100)
Answer is: 425

Another example: 25 x 18
dividing 18 by 4, you get 4 remainder 2.
But that is the same as 4.50
Now move the decimal right two places.
Answer: 450

Another example: 25 x 19
dividing 19 by 4, you get 4 remainder 3.
But that is the same as 4.75
Now move the decimal two places to the right.
Answer is: 475

Here’s the third in my series of multiplication tricks. The first was a trick for multiplying by 5. The second a trick for multiplying by 15, and now this one, a trick for multiplying by 25. Anyone see a pattern?

TRICK #3:

WHAT THE TRICK LETS YOU DO: Quickly multiply numbers by 25.

HOW YOU DO IT: The key to multiplying by 25 is to think about quarters, as in “nickels, dimes, and quarters.”

Since four quarters make a dollar, and a dollar is worth 100 cents, the concept of quarters helps children see that 4 x 25 = 100.

Since four quarters make one dollar, children can see that twice that many quarters, 8, must make two dollars (200 cents). And from that fact children can see that 8 x 25 = 200.

Following this pattern, children can see that twelve quarters make three dollars (300 cents). So 12 x 25 = 300. And so on.

Fine. But how does all of this lead to a multiplication trick?

The trick is this. To multiply a number by 25, divide the number by 4 and then tack two 0s at the end, which is the same as multiplying by 100.

But wait, you protest … what about all of the numbers that are not divisible by 4? Good question! But it turns out that there’s a workaround. You still divide by 4, but now you pay attention to the remainder.

If the remainder is 1, that’s like having 1 extra quarter, an additional 25 cents, so you add 25 to the answer.

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!

Here’s the second in my set of multiplication tricks. (The first was a trick or multiplying by 5.)

TRICK #2:

WHAT THE TRICK LETS YOU DO: Multiply numbers by 15 — FAST!

HOW YOU DO IT: When multiplying a number by 15, simply multiply the number by 10, then add half.

EXAMPLE:15 x 6

6 x 10 = 60

Half of 60 is 30.

60 + 30 = 90

That’s the answer:15 x 6 = 90.

ANOTHER EXAMPLE:15 x 24

24 x 10 = 240

Half of 240 is 120.

240 + 120 = 360

That’s the answer:15 x 24 = 360.

EXAMPLE WITH AN ODD NUMBER:15 x 9

9 x 10 = 90

Half of 90 is 45.

90 + 45 = 135

That’s the answer:15 x 9 = 135.

EXAMPLE WITH A LARGER ODD NUMBER:23 x 15

23 x 10 = 230

Half of 230 is 115.

230 + 115= 345

That’s the answer:15 x 23 = 345.

PRACTICE Set:(Answers below)

15 x 4

15 x 5

15 x 8

15 x 12

15 x 17

15 x 20

15 x 28

ANSWERS Set:

15 x 4=60

15 x 5=75

15 x 8=120

15 x 12=180

15 x 17=255

15 x 20=300

15 x 28=420

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!

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